Overview
Here are the instructions for the homework. I need you to follow everything and make sure you do everything he is asking for also, let me know if you have any other questions before starting.
| Instructions | |
I have extended the due date for problem set #3 to 12 noon on Monday, 10/13/25 in order to make sure that we had enough to time to cover the relevant content in class. As a reminder, you are welcome to work in groups but each person must submit their own assignment written in their own words. Use of AI is prohibited. Presenting another person's work or AI output as your own is an academic integrity violation. Your submission must be in a single Word or PDF file. Your responses should be in the correct order and your submission should be neat and legible, even the handwritten parts. If you copy-paste Excel graphs into another document, the formatting can sometimes get screwed up in the process. To avoid that problem, you can instead do a "special" paste where the graph is pasted as an image. |
PPD 303: Statistics for Policy, Planning, and Development Fall 2025 Mark D. Phillips 1) Combinations and Probability [6 points total]. Statistics and probability play an important role in identifying biased and discriminatory behavior. Suppose a judge hears 25 court cases that are drug-related. 10 of the defendants are Black and 15 are not. a. [1 point] The judge levies a harsh penalty on one Black defendant and a lenient penalty on the remaining 24 defendants. What is the likelihood that this would occur by chance? More precisely, what is that likelihood that a single defendant, randomly selected for a harsh penalty, would be Black? b. [3 points] Suppose instead that the judge levies a harsh penalty on five Black defendants and a lenient penalty on the remaining 20 defendants. What is the likelihood that this would occur by chance? More precisely, what is the likelihood that five defendants, all randomly selected for a harsh penalty, would all be Black? This is harder than part a, but I’m going to help you think through it. Be sure to show your answers to all three of the questions (i, ii, and iii) below. i. If the five defendants are selected at random, then each combination of five defendants is equally likely. How many combinations of five defendants are there among the 25 total defendants? ii. How many combinations of five defendants are there among the 10 Black defendants? iii. Your answer to part i is the total number of combinations of five defendants. Your answer to part ii is the total number of combinations of five defendants where all five are Black. What is the probability of the judge randomly selecting a combination of five defendants where all five are Black? c. [2 points] Suppose you observe that the judge has only levied harsh penalties against Black defendants. The judge denies that race played a role in those decisions and that the racial composition of the defendants receiving harsh penalties is coincidental. A judge who did not make decisions based on race would deny the allegation, but so would a judge who did make decisions based on race. Using your knowledge of probability and your answers to parts a and b, what can you conclude about the judge’s denial? How does your conclusion vary depending on whether the judge had levied a harsh penalty on one vs. five Black defendants? Explain. 2) Normal Distributions [9 points total]. Empirical studies suggest that the height of adult men in the US is approximately normally distributed with a mean of 70 inches and a standard deviation of 3 inches. For parts a-d, first draw the picture of the normal density curve and show the area under the curve that you are looking for. (You should draw a different picture for each of the first four parts. You do not have to draw a picture for part e, though you may still find it useful.) Then calculate the probability using Excel. Please provide the formula that you used in Excel to get the answer. (Don’t just say you used =norm.dist( ) or =norm.inv( ); please also provide the specific arguments that you entered into the function.) Page 2 of 3 a. [2 points] What is the probability that an adult male is less than 72 inches tall? 1 b. [2 points] What is the probability that an adult male is greater than 72 inches tall? c. [2 points] What is the probability that an adult male is within 4.5 inches of the mean (i.e., between 65.5 and 74.5 inches tall)? d. [2 points] What is the probability that an adult male is more than 4.5 inches from the mean? e. [1 point] How tall is an adult male in the 90th percentile of the height distribution? 3) Normal Distributions Using Table A [6 points total]. You will not have access to Excel on the remaining exams and will instead have to rely on Table A to calculate probabilities that involve the normal distribution. Use Table A, not Excel, to solve this problem. As a reminder, empirical studies suggest that the height of adult men in the US is approximately normally distributed with a mean of 70 inches and a standard deviation of 3 inches. a. [1 point] What is the Z-score for an adult male who is 67.9 inches tall? b. [1 point] How would you interpret the Z-score that you calculated in part a? In other words, what does your answer from part a mean? c. [1 point] Using Table A, what is the probability that an adult male is less than 67.9 inches tall? In a sentence or two, explain how you used Table A to determine your answer. d. [1 point] What is the Z-score for an adult male who is 74.5 inches tall? e. [1 point] How would you interpret the Z-score that you calculated in part d? In other words, what does your answer from part d mean? f. [1 point] Using Table A, what is the probability that an adult male is more than 74.5 inches tall? In a sentence or two, explain how you used Table A to determine your answer. 4) Sampling Distributions [6 points total]. Empirical studies suggest that the height of adult men in the US is approximately normally distributed with a mean of 70 inches and a standard deviation of 3 inches. Suppose now that you take a random sample of four (4) adult men. a. [1 point] What is the standard error of the sampling distribution? Calculate the following probabilities using Excel. As a reminder, the sampling distribution is normal whenever the population itself is normally distributed (i.e., you do not have to rely on the Central Limit Theorem for the sampling distribution to be normal). Please provide the formula that you used in Excel to get the answer. (Don’t just say you used =norm.dist( ); please also provide the specific arguments that you entered into the function. You do not need to draw any pictures for this problem.) b. [1 point] What is the probability that the sample mean is less than 72 inches? 2 c. [1 point] What is the probability that the sample mean is greater than 72 inches? d. [1 point] What is the probability that the sample mean is within 4.5 inches of the population mean (i.e., between 65.5 and 74.5 inches)? 1 This is equivalent to asking: “What proportion of adult males are less than 72 inches tall?” 2 This is equivalent to asking: “What proportion of all possible 𝑛𝑛 = 4 random samples have a sample mean less than 72 inches?” Page 3 of 3 e. [1 point] What is the probability that the sample mean is more than 4.5 inches from the population mean? f. [1 point] Compare your answers to 2d (the probability that an adult male is more than 4.5 inches from the mean) vs. 4e (the probability that an 𝑛𝑛 = 4 sample of adult males has a mean more than 4.5 inches from the population mean). Explain how the relative sizes of these two probabilities relate to the Law of Large Numbers. Note: I have asked you to answer 4b-4e using Excel, but you should practice solving these questions using Table A to prepare for the remaining exams.